Elastic constants from microscopic strain fluctuations

Fluctuations of the instantaneous local Lagrangian strain ${\epsilon }_{\mathrm{ij}}(\mathbf{r},\mathbf{t}),$ measured with respect to a static “reference” lattice, are used to obtain accurate estimates of the elastic constants of model solids from atomistic computer simulations. The measured strains are systematically coarse-grained by averaging them within subsystems (of size $L_b)$ of a system (of total size $L)$ in the canonical ensemble. Using a simple finite size scaling theory we predict the behavior of the fluctuations $〈{\epsilon }_{\mathrm{ij}}{\epsilon }_{\mathrm{kl}}〉$ as a function of $L_b/L$ and extract elastic constants of the system in the thermodynamic limit at nonzero temperature. Our method is simple to implement, efficient, and general enough to be able to handle a wide class of model systems, including those with singular potentials without any essential modification. We illustrate the technique by computing isothermal elastic constants of “hard” and “soft” disk triangular solids in two dimensions from Monte Carlo and molecular dynamics simulations. We compare our results with those from earlier simulations and theory.